Calculus: Analytic Geometry and Calculus, with Vectors

11.3 Formulas involving partial derivatives 577

when y = y(x). In this case, x is both a "primary variable" and a

"secondary variable." Differentiating with respect to the "primary

variable" x then gives the formula

(11.341) aF aF dy

ax +ay 7---

from which dy/dx can be calculated when aF/dy 0. This is a fact

involving "implicit functions," the idea being that (11.34) isnot a formula

that gives an explicit formula for y in terms ofx, but (11.34), and per-

haps some further restrictions, may nevertheless imply thaty must be

the unique member or one of the members ofa class of differentiable

functions. The real significance of (11.341) lies in the fact that it often

enables us to obtain a useful formula for dy/dx without undertaking

the sometimes difficult or impossible task of "solving" (11.34) to obtain

a useful formula for Y. Problem 7 at the end of this section gives sub-

stantial information about this matter.

A satisfactory development of our subject

must call attention to the fact that the symbol

au/ax in (11.33) loses its unambiguous meaning

when it is taken out of its context and we are

not sure that the "independent variables" are x

and y. To prove this, we construct Figure

11.35, in which P is supposed to be a point in

the first quadrant. With each such point P, we

may associate the circle through P with center

at the origin and let A be the area of the disk

Figure 11.35

bounded by this circle. If we consider x and y to be the independent

variables that determine P, we obtain the first and then the second of
the formulas

(11.351) a`Q= 2ax.

(x2 + y2), ax

If we consider x and p to be the independent variables, then

(11.352) A = 7rp2

M

= 0.

TX_

If we consider x and 0 to be the independent variables, then

(11.353)

-2 a A 1_...

= 7r cost p, ax = cost (A.

Since the symbol all/ax has different values in different contexts, the
symbol must be discarded or embellished when there is no clear specifica-
tion of the identities of the independent variables. The first of the
symbols

ail l aA

ax axLO,v=o